The Old Testament laws are categorized in three groups: the civil, the priestly, and the moral. The civil laws must be understood in the context of a theocracy. Though the Jewish nation in the Old Testament was often headed by a king, it was a theocratic system with the Scriptures as a guide to the nation. Those laws that fall under this category are not applicable today because we are not under a theocracy.
The priestly laws dealing with the Levitical and Aaronic priesthoods, were representative of the future and true High Priest Jesus who offered Himself as a sacrifice on the cross. Since Jesus fulfilled the priestly laws, they are no longer necessary to be followed and are not now applicable.
The moral laws, on the other hand, are not abolished. Because the moral laws are based upon the character of God. Since God's holy character does not change, the moral laws do not change either. Therefore, the moral laws are still in effect.
In the New Testament we do not see a reestablishment of the civil or priestly laws. But we do see a reestablishment of the moral law. This is why we see New Testament condemnation of homosexuality as a sin but not with the associated death penalty.